Linear Instability of Sasaki Einstein and nearly parallel ${\rm G}_2$ manifolds

TL;DR

This paper investigates the linear stability of Einstein metrics on Sasaki Einstein and nearly parallel G2 manifolds, establishing instability conditions based on Betti numbers and analyzing specific examples like the Berger space.

Contribution

It provides new criteria for linear instability of Einstein metrics on these manifolds based on Betti numbers and examines specific examples such as the Berger space.

Findings

Linear instability occurs when the second Betti number is positive for Sasaki Einstein manifolds.

Nearly parallel G2 manifolds with positive third Betti number are linearly unstable.

The Berger space SO(5)/SO(3)_{irr} is shown to be linearly unstable.

Abstract

In this article we study the stability problem for the Einstein metrics on Sasaki Einstein and on complete nearly parallel ${\rm G}_2$ manifolds. In the Sasaki case we show linear instability if the second Betti number is positive. Similarly we prove that nearly parallel $\rm G_2$ manifolds with positive third Betti number are linearly unstable. Moreover, we prove linear instability for the Berger space ${\rm SO}(5)/{\rm SO}(3)_{irr} $ which is a $7$-dimensional homology sphere with a proper nearly parallel ${\rm G}_2$ structure.